Analytic properties of an orthogonal Fourier–Jacobi Dirichlet series

We investigate the analytic properties of a Dirichlet series involving the Fourier–Jacobi coefficients of two cusp forms for orthogonal groups of signature $$(2,n+2)$$ ( 2 , n + 2 ) . Using an orthogonal Eisenstein series of Klingen type, we obtain an integral representation for this Dirichlet series. In the case when the corresponding lattice has only one 1-dimensional cusp, we rewrite this Eisenstein series in the form of an Epstein zeta function. If additionally 4 n 4 ∣ n , we deduce a theta correspondence between this Eisenstein series and a Siegel Eisenstein series for the symplectic group of degree 2. We obtain, in this way, the meromorphic continuation of the Dirichlet series to C C as a corollary. In the case of the E₈ E 8 lattice, we are able to further deduce a precise functional equation for the Dirichlet series.